Probability

Question

Answer 1

Probability of drawing one ball = $\frac{1}{20}$

$X_{i}$ (ball number)	$1$	$2$	$3$	....	$20$
$P\left ( X_{i} \right )$ (Probabilitiy of drawing that ball)	$\frac{1}{20}$	$\frac{1}{20}$	$\frac{1}{20}$	....	$\frac{1}{20}$

$E(X_{i})$ = $\sum_{i=1}^{20} X_{i}*P(X_{i})$

$E(X_{i}) = 1*\frac{1}{20} + 2* \frac{1}{20} + 3*\frac{1}{20} + .... + 20*\frac{1}{20}$

$E(X_{i}) = \frac{1}{20} \left ( 1+ 2 + 3 + .... + 20 \right )$

$E(X_{i}) = \frac{1}{20} \left ( \frac{20*21}{2}\right )$

$E(X_{i}) = \frac{21}{2}$

Required expectation = $E(X_{1}) + E(X_{2}) + E(X_{3}) + .... + E(X_{8})$ 

                                       = $8*\frac{21}{2}$

                                       = $84$

Comments

@Shubhanshu, even if you do that way, you'll get the same answer. I have simply found the expected value for 20 numbers first then multiplied by 8, even your approach is correct

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@joshi_nitish try for a smaller set like numbers between 1 to 5 and k = 3, you'll get correct answer.

https://math.stackexchange.com/questions/1712491/expected-value-of-sum-of-distinct-random-integers

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Thanks, just_bhavana previously I am doing calculation mistake.